Before we proceed with optimization for inhomogeneous JJ chains,
it is useful to see what can be achieved in the homogeneous case,
for future reference. For the problem
(REF )–(REF ), with all
\(\mathcal {A}_n=\mathcal {A}\) , we have only two variables,
\(\mathcal {A}\)  and \(N\) . Constraint (REF )
fixes \(\mathcal {A}=N/N_0\) , \(L_n=L_\mathrm {max}N_0/N\) ,
\(C_n=C_\mathrm {min}N/N_0\) ,
\(C^\mathrm {g}_n=C^\mathrm {g}_\mathrm {min}\,g(N/N_0)\) .
It is convenient to denote the first mode frequency for
this homogeneous chain by \(\Omega _N\) . It is given by [1]}
\(\Omega _N^2={}&{}\frac{1}{LC}\,\frac{1-\cos [\pi /(N+1)]}{1-\cos [\pi /(N+1)]+C^\mathrm {g}/(2C)}\approx \nonumber \\\approx {}&{}\frac{(L_\mathrm {max}C_\mathrm {min})^{-1}}{1+(C^\mathrm {g}_\mathrm {min}/C_\mathrm {min})(N_0/\pi )^2[x/g(x)]^2}\) 
